https://file.aiquickdraw.com/mathbot/file/17264509318170gnz80b5.jpg

Let's analyze the problem given in the image.

The function provided is:
\[
f(x) = 18 - |x|
\]
We are asked to determine whether this function is differentiable at \(x = 0\), by considering the definition of the derivative:
\[
f'(0) = \lim_{h \to 0} \frac{f(0 + h) - f(0)}{h}
\]

### Step 1: Evaluate \( f(0) \)
Given \( f(x) = 18 - |x| \), we have:
\[
f(0) = 18 - |0| = 18
\]

### Step 2: Evaluate \( f(0 + h) \)
When \( h > 0 \), we have \( f(0 + h) = 18 - h \), and when \( h < 0 \), we have \( f(0 + h) = 18 + h \) (since \( |h| = -h \) for negative \( h \)).

### Step 3: Compute the left-hand and right-hand limits

- **Right-hand limit** (\( h > 0 \)):
\[
\lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^+} \frac{(18 - h) - 18}{h} = \lim_{h \to 0^+} \frac{-h}{h} = -1
\]

- **Left-hand limit** (\( h < 0 \)):
\[
\lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^-} \frac{(18 + h) - 18}{h} = \lim_{h \to 0^-} \frac{h}{h} = 1
\]

### Step 4: Conclusion
Since the left-hand and right-hand limits are not equal, the derivative does not exist at \( x = 0 \), which means the function is **not differentiable** at \( x = 0 \).

### Correct Answer:
The correct answer is **B**: "The function \( f \) is not differentiable at \( x = 0 \) because the left- and right-hand limits of the difference quotient exist at \( x = 0 \), but are not equal."

Would you like any further details or have any questions about this? 

Here are some related questions:
1. What are the conditions for a function to be differentiable at a point?
2. How does differentiability relate to continuity?
3. Can you give examples of functions that are continuous but not differentiable?
4. How does the absolute value function affect differentiability at certain points?
5. What is the geometrical interpretation of a function not being differentiable at a point?

**Tip**: A sharp corner or cusp in a graph often indicates that the function is not differentiable at that point.

Determine whether the function f(x) = 18 - |x| is differentiable at x = 0 by considering the limit definition of the derivative.

Explore whether the function f(x) = 18 - |x| is differentiable at x = 0 using the limit definition of the derivative. Learn about the importance of matching left-hand and right-hand limits to determine differentiability, ideal for Grades 11-12 students.

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